Question

The lobes of which \(d\) orbitals point directly between the ligands in (a) octahedral geometry, (b) tetrahedral geometry?

Short answer

In (a) octahedral geometry, the lobes of the \(d_{xy}\), \(d_{yz}\), and \(d_{xz}\) orbitals point directly between the ligands. In (b) tetrahedral geometry, the lobes of the \(d_{z^2}\) and \(d_{x^2-y^2}\) orbitals point directly between the ligands.

Step-by-step solution

Understanding the d Orbitals

The d orbitals have a total of 5 different shapes. They can be represented as: 1. \(d_{xy}\): lobes lie in xy-plane, between x and y axes 2. \(d_{yz}\): lobes lie in yz-plane, between y and z axes 3. \(d_{xz}\): lobes lie in xz-plane, between x and z axes 4. \(d_{z^2}\): lobes lie along the z-axis, with a torus shape in the xy-plane 5. \(d_{x^2-y^2}\): lobes lie along the x and y axes

Understanding Octahedral Geometry

In octahedral geometry, six ligands are arranged around the central atom. The ligands form opposite pairs along each of the three axes (x, y, and z) in a symmetric manner.

Identifying the d Orbitals for Octahedral Geometry

As the ligands are arranged along the axes in octahedral geometry, we have to identify the d orbitals that point between the ligands. In this case, the \(d_{xy}\), \(d_{yz}\), and \(d_{xz}\) orbitals have lobes that point directly between the ligands, because their lobes lie between the axes.

Understanding Tetrahedral Geometry

In tetrahedral geometry, four ligands are arranged around the central atom. The ligands are placed at the vertices of a regular tetrahedron, which results in each ligand occupying a position between any two axes.

Identifying the d Orbitals for Tetrahedral Geometry

Since the ligands in tetrahedral geometry occupy positions between any two axes, the d orbitals with lobes pointing directly between the ligands are the ones with lobes along the axes. In this case, the \(d_{z^2}\) and \(d_{x^2-y^2}\) orbitals have lobes that point directly between the ligands.

Conclusion

In (a) octahedral geometry, the lobes of the \(d_{xy}\), \(d_{yz}\), and \(d_{xz}\) orbitals point directly between the ligands. In (b) tetrahedral geometry, the lobes of the \(d_{z^2}\) and \(d_{x^2-y^2}\) orbitals point directly between the ligands.

Key concepts

Octahedral Geometry

Octahedral geometry refers to a specific arrangement of atoms or ligands around a central atom, forming an octahedron. In this configuration, there are six ligands, one at the center of each face of the octahedron, effectively creating three intersecting perpendicular axes: the x, y, and z axes.

An octahedral structure exhibits high symmetry and is common in coordination chemistry. An example of this is the complexes formed by transition metals, where six ligands accompany the metal center.

Due to the arrangement of the ligands along the axes, certain d orbitals will point directly between the ligands. Specifically, the lobes of the \(d_{xy}\), \(d_{yz}\), and \(d_{xz}\) orbitals are perfectly aligned to fall between these axes, as they lie within the respective planes but do not align directly with the axes themselves. This results in minimal interaction, making these orbitals the most stable in an octahedral field.

The octahedral geometry plays a crucial role in determining the electronic structure and properties of a complex, such as its color, magnetic properties, and reactivity.

Tetrahedral Geometry

Tetrahedral geometry is an arrangement where four ligands are symmetrically positioned around a central atom at the vertices of a tetrahedron.

Unlike octahedral geometry, here the ligands do not align with the central axes but rather are located in positions between any two axes, creating equal bond angles of approximately 109.5 degrees. This results in a less symmetric structure compared to the octahedral arrangement.

In the context of d orbitals, the \(d_{z^2}\) and \(d_{x^2-y^2}\) orbitals are significant in tetrahedral geometry. These orbitals have lobes pointing more directly along the axes, unlike the others which lie more between or at angles to the axes. Hence, the \(d_{z^2}\) and \(d_{x^2-y^2}\) are more involved in the direct interaction with the ligands.

Tetrahedral structures are often seen in smaller metal ions with fewer coordination numbers, and the distorted symmetry can lead to different spectroscopic and electronic properties compared to octahedral complexes.

Ligand Coordination

Ligand coordination refers to the specific interaction between a central metal atom and surrounding molecules or ions called ligands. These can donate electrons to form coordinate covalent bonds, creating stable and complex structures.

In a coordination complex, the geometry and properties of the complex are significantly influenced by the type and number of ligands that coordinate to the metal.

Ligands affect the spatial orientation of d orbitals, influencing their energy levels and distribution. For instance, in octahedral coordination, the coordination changes the energies of the d orbitals, splitting them into groups. This is crucial for understanding the structure and reactivity of transition metal complexes.

Understanding how ligands coordinate with metal centers helps predict many chemical and physical properties, including color, magnetism, and stability. These properties are vital in fields like bioinorganic chemistry, catalysis, and materials science, where precise control over coordination is essential.